The main difficulty that Quantum Chemistry, as a discipline, has found in the main stream chemical community is rooted on the definition domain of the molecular wave function: the whole 3D space. This is the famous Hilbert space problem, and prevents that properties depending on a localized region of space be rigourously quantum mechanically defined. On the other hand, Chemistry is inseparable from local concepts, and much work has been done along the years to overcome this difficulty. None of the large number of approaches that have been proposed, except that of Prof. Bader ( Bader 1990), is fully coherent from the theoretical point of view.

Bader's main concept is a theorem
proving that non relativistic quantum mechanics is applicable to open regions
of the 3D space if these regions are bounded by surfaces whose flux of
the gradient of the electronic density vanishes. The existence of these
surfaces or separatrices
is assured for any well behaved scalar density ---as the electronic
density is--- and allows us to
partition the physical space into non-overlapping regions. The latter contain,
in general, only one nucleus. The problem associated to the practical
determination of the position of the separatrices is intimately related to the
topology of the gradient field and, particularly, to the position and
structure of the zero gradient points or ** Critical Points ** (CP's).

Four non-degenerate kinds of CP's are possible in 3D space: Maxima, minima, first kind saddles, and second kind saddles, according to the number of positive eigenvalues of the Hessian matrix of the electronic density scalar field at the point considered (0,1,2, and 3, respectively). They are also known in the AIM jargon as nuclear, cage, bond, and ring points. This chemical association will turn clear in brief.

A flavour of the localization of critical points in an actual crystal may be grasped from the following figure, where we present the computed isoelectronic density plot along with the critical points of the electron density in the [100] plane of the rock-salt (B1) phase of LiCl at the theoretical equilibrium distance.

It is also important in what follows to recall the concept of field lines or
gradient following trajectories of a vector field. Every field line starts
at, and ends up at a CP. Nuclei, or maxima, are atractors of the field lines
as a 3D set of trajectories end at a given nuclei. Cage point, on the contrary,
are sources, or repulsors. The set of points of the space whose field lines
share a same end-point (or start-point)
CP is called the basin of attraction (or repulsion) of the CP. In the case of a
maximum, the union of the nuclear point plus its basin of attraction is
identified with the chemical concept of atom. When considering a bond point,
the two lines originating from it have different nuclear attractors. We say
that those nuclei are bonded. The network formed by the nuclei and their bonds
is a connected graph, the * molecular graph*. When the molecular graph
displays a cycle, the system is said to have a ring. It is found that a
ring of bonds is also associated with a ring point, located somewhere in
between the ringed nuclei. Finally, a set of non-coplanar rings may create a
cavity holding a cage point in its interior.

This redefinition of our most deeply rooted chemical concepts would be of no value if the partition of space induced by the topology of the electronic density did not allow for the calculation of local properties. In fact, any quantum mechanical expected value is found to be the addition of the local expected values over the basins of all the topological atoms. In this way, any property is additively partitioned into atomic contributions. Group properties are immediately obtained from atomic ones in the very moment we associate a group with a set of atoms. At this stage, the theory barely starts to show its magnificience.

The periodicity of a crystalline material changes a little bit this schematic panorama. First of all, the electronic density space domain is now homeomorphic to a 3-torus. In fact, this condition forces that every atomic basin be finite and, therefore, that the concept of atomic volume, such an important one in solid state physics, be rigourously defined. It may be easily shown ( Pendás 95) that space group symmetry greatly restrict the possible positions and types of CP's, and that the total number of CP's of every kind must fulfil Morse's relations:

n - b + r - c = 0

and

n>=1, b>=3, r>=3, c>=1, n,b,r, and c standing for the number of maxima, bond, ring, and cage points, respectively.

Periodicity also induces a partition of space into smaller regions than those
originally proposed in the AIM theory. If we define a ** primary bundle **
(PB)
as the set of trajectories starting at a given minimum and ending at a given
maximum, the bundle is the minimum region of space surrounded by a zero-flux
surface. It may be shown that a new homeomorphism is induced between a PB and
a convex polyhedron with 2n+2 vertices, 2n faces, and 4n edges.
Moreover, every finite AIM topological object must be a finite union of PB's.
One grouping recipe is that of gathering all PB's sharing the same nucleus.
The object just defined coincides with Bader's atomic basin.
**Atomic shapes**,
therefore, are also homeomorphic to **polyhedra**.
The mapping is as follows:
cage points are vertices, bond points faces, and ring points edges. Another
way to group PB's is just the opposite of the one just considered. Join all
bundles sharing the same cage. The object is called repulsion basin of a cage,
and the associated polyhedron, repulsion polyhedron.

Once these basins and polyhedra have been defined, a familiar image gets its
shape. to every atom we associate an atomic polyhedron with as many faces as
bonds connecting this atom to the lattice. Repulsion polyhedra are the
equivalent to coordination polyhedra. To help visualize this objects, a whole
series of approximants to their actual structure may be constructed.
For example, if we remain with just the topology of the different polyhedra,
the ** proximity or Voronoi polyhedra ** (PP) (Wigner-Seitz cells of a hypothetical
Bravais lattice containing one point per nuclear position) should be, in most
cases, good approximations to the actual atomic polyhedra. Since there exist
powerful algorithms to obtain PP's from the coordinates of a lattice,
a relatively quick picture of atomic shapes and coordination polyhedra
may be obtained easily. More rigourous shapes obtained by triangulation
techniques will be commented below. The following figures show the PP and the
coordination polyhedra in the LiCl case.

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